In this paper one presents the extension of the transient analysis of the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend on the state of the process. We improve previously known ergodicity and truncation bounds for this class of processes which were known only for the case when transitions from the origin decay exponentially (other intensities must have unique uniform upper bound). We show how the bounds can be obtained the decay rate is slower than exponential. Numerical results are also provided.